Special limits. Game on.

Special limits is a grade 12 math skill: evaluate special limits, including limits at infinity and limits of indeterminate forms that define key constants and derivatives. Below are 3 practice questions with answers and step-by-step explanations, drawn from the 5 special limits problems our math games drill.

5 questions in the bank
Kickoff

3 plays. No signup.

Warm-upeasy

Evaluate: lim⁡x→0ex−1x\displaystyle\lim_{x \to 0} \dfrac{e^x - 1}{x}.

This is a standard memorized limit: lim⁡x→0ex−1x=1\displaystyle\lim_{x \to 0} \dfrac{e^x - 1}{x} = 1.

Mid-gameeasy

Evaluate: lim⁡x→0tan⁡xx\displaystyle\lim_{x \to 0} \dfrac{\tan x}{x}.

tan⁡xx=sin⁡xx⋅1cos⁡x→1⋅1=1\dfrac{\tan x}{x} = \dfrac{\sin x}{x} \cdot \dfrac{1}{\cos x} \to 1 \cdot 1 = 1.

Buzzer beatereasy

Evaluate: lim⁡x→01−cos⁡xx2\displaystyle\lim_{x \to 0} \dfrac{1 - \cos x}{x^2}.

Using the approximation 1−cos⁡x≈x221 - \cos x \approx \dfrac{x^2}{2} near 00: 1−cos⁡xx2→12\dfrac{1 - \cos x}{x^2} \to \dfrac{1}{2}.

Overtime

2 more. Played, not assigned.

The rest of the bank lives inside live head-to-head matches — the placement test finds the right starting difficulty.

Batting Cage game coverBatting CagePlay →Blossom Leap game coverBlossom LeapPlay →Buzzer Beater game coverBuzzer BeaterPlay →Minecart Junction game coverMinecart JunctionPlay →
More Grade 12 skills
Polar and rectangular conversionStandard antiderivativesMaclaurin seriesStandard derivativesInverse trig domain and rangeDegree-radian conversionTrig identitiesConvergence testsAll Grade 12 practice →

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